Question

The vertices of a triangle are at (0,0), (6,6), and (9,3) . What is the volume, in cubic units, of the figure created by rotating the triangle about y=x?

Answer 1

Answer:

volume of the figure after rotation=56.52 cubic units.

Step-by-step explanation:

After rotating the figure around the line y=x, we will see that the coordinates (x,y) change to the coordinates (y,x).

Hence the coordinates of the triangle (0,0),(6,6) and (9,3) changes to (0,0),(6,6) and (3,9).

we will get a cone on rotating the triangle across y=x.

The radius of cone(r)=3 units.

height of cone(h)=6 units.

volume of the cone=  $\dfrac{1}{3} \pi r^2h$.

Hence,  volume of cone= 56.52 cubic units.

Answer 2

Answer:

The triangle rotated around y=x forms a cone with a volume of 36π√(2), or about 159.94.

Step-by-step explanation:

Use the distance formula or the Pythagorean theorem to find that the distance between (6,6), and (9,3) is 3√(2). I like the P.T., so I draw a right triangle with legs of 3&3 so the hypotenuse is the distance between the points. This is the radius of the cone. r²=3²+3²=18, so r=√(18)=√(9·2)=3√(2)

So r=3√(2). We also know that r²=9·2=18.

Use the distance formula or Pythagorean theorem to find the distance between (0,0) and (6,6) to find the height of the cone. The Pythagorean theorem gives us a triangle with legs = 6 and 6. Therefore, 6²+6²=h², or 72=h². √(72)=√(36·2·)=6√(2). So h=6√(2).

The volume of a cone is 1/3 the volume of the cylinder with the same base and height, which is just the area of a circle times the height. So, (1/3)(πr²)·h.

(1/3)(18π)(6√2)=6π·6√2=36π√2.

I hope that helps! This is the answer the NC testing board got for this question, so I feel good about it.